Solution of Triangles: Circles Circumscribing or Inscribing a Regular Polygon

When dealing with regular polygons, two important circles can be defined: the circumscribed circle and the inscribed circle. The circumscribed circle, or circumcircle, is the circle that passes through all the vertices of the polygon, while the inscribed circle, or incircle, is the circle that is tangent to each side of the polygon.

Circumscribed Circle (Circumcircle)

For a regular polygon with $n$ sides (an $n$-gon), the radius of the circumscribed circle, often denoted as $R$, is the distance from the center of the polygon to any of its vertices.

Formula for Circumradius ($R$)

The formula for the circumradius $R$ of a regular $n$-gon with side length $a$ is given by:

$$ R = \frac{a}{2\sin\left(\frac{\pi}{n}\right)} $$

Properties of the Circumcircle

• The center of the circumcircle is also the center of the regular polygon.
• The circumradius is the radius of the regular polygon.
• All the vertices of the polygon lie on the circumference of the circumcircle.

Inscribed Circle (Incircle)

The inscribed circle is the largest circle that fits inside the regular polygon, touching all its sides. The radius of the inscribed circle is called the inradius, often denoted as $r$.

Formula for Inradius ($r$)

The formula for the inradius $r$ of a regular $n$-gon with side length $a$ is given by:

$$ r = \frac{a}{2\tan\left(\frac{\pi}{n}\right)} $$

Properties of the Incircle

• The center of the incircle is also the center of the regular polygon.
• The inradius is the apothem of the regular polygon.
• Each side of the polygon is tangent to the incircle.

Table of Differences and Important Points

Property	Circumcircle	Incircle
Definition	Circle passing through all vertices	Circle tangent to all sides
Radius	Circumradius ($R$)	Inradius ($r$)
Formula for Radius	$R = \frac{a}{2\sin\left(\frac{\pi}{n}\right)}$	$r = \frac{a}{2\tan\left(\frac{\pi}{n}\right)}$
Relation to Polygon	Radius equals the distance from center to a vertex	Radius equals the apothem (distance from center to a side)
Center	Same as the center of the polygon	Same as the center of the polygon

Examples

Example 1: Circumradius of a Hexagon

Find the circumradius of a regular hexagon with a side length of 10 units.

Solution:

For a hexagon, $n = 6$. Using the formula for the circumradius:

$$ R = \frac{a}{2\sin\left(\frac{\pi}{n}\right)} = \frac{10}{2\sin\left(\frac{\pi}{6}\right)} = \frac{10}{2 \cdot \frac{1}{2}} = 10 $$

The circumradius of the hexagon is 10 units.

Example 2: Inradius of a Pentagon

Find the inradius of a regular pentagon with a side length of 8 units.

Solution:

For a pentagon, $n = 5$. Using the formula for the inradius:

$$ r = \frac{a}{2\tan\left(\frac{\pi}{n}\right)} = \frac{8}{2\tan\left(\frac{\pi}{5}\right)} \approx \frac{8}{2 \cdot 0.7265} \approx 5.51 $$

The inradius of the pentagon is approximately 5.51 units.

Understanding the relationship between regular polygons and their circumscribed and inscribed circles is crucial in various fields of mathematics and engineering, such as geometry, trigonometry, and design. The formulas provided here are essential tools for solving problems involving these geometric figures.